The mean filed Gross-Pitaevskii (GP) equation is not sufficient for the droplet; we need to consider the higher-order correction, popularly known as LHY [1]} correction term.
The well established coupled GP equations for the 2D droplets is given by [2]}

\(i\frac{\partial \psi _1 }{\partial t}=\left[-\frac{\nabla ^2}{2} + g(|\psi _1|^2 - |\psi _2|^2) + \frac{g^2}{4\pi } \; \rho ln(\rho ) + V(\vec{r}) \right]\psi _1 \nonumber \\i\frac{\partial \psi _2 }{\partial t}=\left[-\frac{\nabla ^2}{2} + g(|\psi _2|^2 - |\psi _1|^2) + \frac{g^2}{4\pi } \; \rho ln(\rho ) + V(\vec{r}) \right]\psi _2.\) 
